Department/College/UnitPhysical & Biological Sciences Division

TitleDistinguished Professor

AffiliationFaculty

Web Sitehttps://tromba.sites.ucsc.edu

Primary Office LocationMcHenry Library
McHenry Room #4151

Mail StopMathematics Department

Biography, Education and Training

B.S., Cornell University
M.A., Ph.D., Princeton University

Summary of Expertise

Tony Tromba‘s main body of research is on minimal surfaces, a subject about which he has written several Grundlehren volumes. These are surfaces that minimize area among all surfaces bounding a given curve in space (think of soap films bounding wires in space). He developed a Morse Theory and Degree Theory for such surfaces paralleling Morse’s theory for geodesics, results Morse failed to achieve, theories which were consequently honored by an invitation to deliver an address to The International Congress of Mathematicians. He also developed a new approach to Teichmüller Theory and applied this to solve one of the main open questions in the field, namely to compute the sectional curvature of Teichmüller Space with respect to its Weil Petersson Metric. Most recently he has found new positive results for Hilbert's famous 19th problem in the Calculus of Variations. 

Teaching Interests

Tony has been instrumental in developing new and better ways to teach calculus, writing (with Jerry Marsden) one of the outstanding worldwide textbooks on Vector Calculus, a textbook now in print (in five languages) for half a century. Together with his colleague Frank Bäuerle he developed the UC-wide online four course calculus sequence for STEM, the only courses (in any subject) in UC's 157 year history to be offered for credit towards major requirements on all UC undergraduate campuses.

Selected Publications

• On the Hölder Regularity of All Extrema in Hilbert's 19th Problem  (with F. Tomi);  Advances in The Calculus of Variations  2022, (17) Issue 1
• Morse Theory and The Calculus of Variations; Advances in The Calculus of Variations 2023 (16), issue 4
• Morse Theory and Hilbert's 19th Problem (with F. Tomi); Calculus of Variations and Partial Differential Equations, (58) September 2019, 172-184
• L. Andersson, V. Moncrief, and A. Tromba: On the global evolution problem in 2+1 gravit, J. Geometry and Physics 23 (1997), 191-205.
• A. Tromba: On a natural affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil-Petersson metric. Manu-scripta Math 56 (1996), 475-497.
• F. Tomi and A. Tromba: The index theorem for minimal surfaces of higher genus, Mem. Amer. Math Soc 117, No. 560 (1995).
• F. Tomi and A. Tromba: Existence theorems for minimal surfaces of non-zero genus spanning a contour. Mem. Amer. Math. Soc. 71, no. 382 (1988).
• A. Tromba: A general approach to Morse theory. J. Differential Geometry 12:47Ð85. 1 (1977)

Recent Books

•  Regularity of Minimal Surfaces (with U. Dierkes and S. Hildebrandt); Grundllehren der Mathematischen Wissenschaften, (Springer Verlag)
•  Global Analysis of Minimal Surfaces(with U. Dierkes and S. Hildebrandt); Grundllehren der Mathematischen Wissenschaften, (Springer Verlag)